Disturbance Regimes and Life-History Evolution

vol. 157, no. 5 the american naturalist may 2001

David A. Lytle*

Department of Entomology and Field of Ecology & Evolutionary 1961; MacArthur and Levins 1967; Huston 1979; Chesson Biology, Cornell University, Ithaca, New York 14853 1994; Lavorel and Chesson 1995), eliminate nonnative taxa (Meffe 1984; Minckley and Meffe 1987; Closs and Lake Submitted June 14, 2000; Accepted December 26, 2000 1996), facilitate invasive taxa (McEvoy et al. 1993), and alter community food web structure (Wootton 1998). Al- though it seems reasonable that strong ecological forces acting within populations could influence the evolution of abstract: Disturbance regimes are ecologically important, but life-history strategies or morphologies, variation in dis- many of their evolutionary consequences are poorly understood. A turbance timing, predictability, frequency, and severity can model is developed here that combines the within- and among- season dynamics of disturbances with evolutionary life-history the- make it difficult to predict the sign and strength of selec- ory. “Disturbance regime” is defined in terms of disturbance timing, tion. Several studies (Harper 1977; Lacey et al. 1983; Ven- frequency, predictability, and severity. The model predicts the optimal able and Brown 1988; Turner et al. 1998) have suggested body size and time at which organisms should abandon a distur- that the frequency of disturbances relative to an organism’s bance-prone growth habitat by maturing and moving to a distur- life span may be evolutionarily important. While it is in- bance-free, nongrowth habitat. The effects of both coarse-grained tuitive that organisms may not adapt to phenomena that (those affecting the entire population synchronously) and fine- grained disturbances (those occurring in a patch dynamics setting) are unlikely to occur during their life spans (e.g., volca- are explored. Several predictions are congruent with previous theory. noes, large fires, big floods, or storms [Turner et al. 1998]), Infrequent or temporally unpredictable disturbances should have lit- it is not clear how frequently disturbances must recur in tle effect on the evolution of life-history strategies, even though they order to elicit evolutionary responses. may cause high mortality. Similar to seasonal time constraints on From an evolutionary perspective, disturbances can be reproduction, disturbance regimes can synchronize metamorphosis categorized as either fine-grained events that affect only a within a population, resulting in a seasonal decline in body size at portion of the population at a time or coarse-grained maturity. Other model predictions are novel. When disturbances cause high mortality, coarse-grained disturbances have a much events that affect the entire population simultaneously stronger effect on life-history strategies than fine-grained distur- (Iwasa and Levin 1995). Fine-grained disturbances (the bances, suggesting that population structure (relative to the scale of “patch dynamics” perspective of Pickett and White [1985]) disturbance) plays a critical evolutionary role when disturbances are include gap formation in forest canopies (Runkle 2000), severe. When within-population variance in juvenile body size is flash floods (Lytle 2000a), and scouring of marine benthos high, two consecutive seasonal declines in body size at maturity can (Airoldi 2000). Coarse-grained disturbances include inter- occur, the first associated with disturbance regime and the second associated with seasonal time constraints. annual variability in growing season length or annual rainfall (Philippi 1993; Danforth 1999), as well as disturbances Keywords: body size, timing of metamorphosis, patch dynamics, with large areal coverage, such as hurricanes (Turner et state-dependent strategy, geometric mean fitness, arithmetic mean al. 1998). The spatial scale of a disturbance relative to the fitness. spatial distribution of the population is important because it determines how fitness should be estimated in models While the ecological effects of disturbances have been rel- of life-history evolution. When disturbances occur syn- atively well studied, the evolutionary consequences of dis- chronously over the entire population, as with coarse- turbances are less understood. Ecologically, disturbances grained disturbances, the geometric mean of reproductive can mediate the coexistence of competitors (Hutchinson success over multiple seasons is the correct measure of fitness (Cohen 1966; Gillespie 1977). If the population

* Present address: Department of Entomology, University of Arizona, Tucson, occurs across many habitat patches that experience dis- Arizona 85721; e-mail: [email protected]. turbances at different times, as with fine-grained distur- Am. Nat. 2001. Vol. 157, pp. 525–536. ᭧ 2001 by The University of Chicago. bances, and the breeding population consists of individuals 0003-0147/2001/15705-0005$03.00. All rights reserved. pooled from these patches, the arithmetic mean is appro- 526 The American Naturalist priate. Thus, evolutionary models incorporating fine- and size at and timing of maturity? How frequently and pre- coarse-grained disturbances are inherently different. In dictably must disturbances recur to affect the evolution of practice, models can sometimes be modified to account these traits? How does population structure influence the for one case or the other (Iwasa and Levin 1995; see evolutionary response to disturbance? Used in this way, below). this disturbance model may be useful for determining Much of the theory concerning how disturbances affect when ecologically important disturbance regimes also have life-history evolution has focused on coarse-grained dis- evolutionary consequences. turbances. Building on the theory of Cohen (1966, 1970, 1971), models have been developed to explore how re- Disturbance Model sources in plants are allotted to growth versus reproduction when the length of the growing season varies across The following model explores how among-season varia- years (King and Roughgarden 1982a, 1982b; Kozłowski bility in disturbance regime (sensu Cohen 1966 and related and Weigert 1986, 1987). For organisms that produce dia- papers) and within-season disturbance dynamics (based pausing seeds or eggs, bet-hedging models predict that on Ludwig and Rowe 1990; Rowe and Ludwig 1991) affect among-year environmental variability may favor repro- life-history evolution. In this model, disturbances affect ductive strategies where only a fraction of offspring ger- individual fitness directly via mortality and indirectly by minate or hatch in a given season (Venable and Lawlor causing mortality in offspring. The model is based on the 1980; Ellner 1985a, 1985b; Bradford and Roff 1993, 1997; following life cycle: juveniles grow in a particular habitat Sasaki and Ellner 1995). In each of these models, the “dis- where they risk mortality from disturbances; at time T, turbance” is the occurrence of an unfavorable physical juveniles stop growing and begin metamorphosis for a environment in a particular year, and the life-history strat- fixed time period; at time TE, nongrowing adults move to egy that maximizes the geometric mean of reproductive a second habitat that is free from disturbance; at time TR, output over many years has the highest fitness. The dis- adults reproduce by placing offspring back in the disturbed turbance does not need to be abiotic, however. Hairston habitat. Thus, juveniles face a trade-off between growth and Munns (1984) used a similar approach to model how and disturbance mortality. Because the risk of disturbance among-year variability in the onset of severe fish predation changes during the season, the model seeks the optimal affected the optimal time for copepods to begin producing body size, W, and time, T, at which juveniles should stop fish-resistant diapausing eggs. growing and mature into the reproductive stage. Most of these coarse-grained models focus on environmental variability among years, but many disturbance dy- Disturbances and Survivorship namics occur within years. Relevant parameters include the frequency (expected number of disturbances per sea- The disturbance regime consists of the timing, predicta- son), severity (expected mortality from a single distur- bility, frequency, and severity of disturbances. Survivorship bance), timing (when disturbances occur during a season), is a function of the time spent in this disturbance regime. and predictability (variance in within-season timing) of The probability of an individual surviving i disturbances p Ϫ i disturbances (Pickett and White 1985; Richter et al. 1996). before adulthood isSi (1 l) , where l is the proba- Although seasonal timing and predictability are implicit bility of mortality from a single disturbance event (dis- in many of the coarse-grained models, they assume that turbance severity). Assuming that disturbance events occur only one disturbance occurs per season (frequency p 1 ). independently according to a Poisson distribution, the Some types of disturbance, however, occur multiple times probability of i disturbances occurring from some time t per season or not at all, for example, flash floods (John to adulthood at time TE is 1964; Grimm and Fisher 1989) and the drying and refilling i Ϫu1 of temporary ponds (Semlitsch and Wilber 1988; Newman ue1 P p ,(1) 1989). Thus, a parameter that specifies within-season fre- i i! quency is needed to adequately model these kinds of p ∫TE disturbances. whereu1 t g(t)dt , a time-inhomogeneous Poisson rate The goal of this article is to develop a general distur- parameter. The frequency of disturbances, such as thun- bance model that combines the timing, frequency, severity, derstorms and flash floods, conforms to a Poisson distri- and predictability of disturbances (both fine and coarse bution (Fogel and Duckstein 1969; Lytle 2000b). The func- grained) with evolutionary life-history theory. This the- tion g(t) describes the timing of disturbances throughout oretical framework allows the investigation of several ques- the season; it is assumed to have a single maximum. In tions. How do disturbance regimes affect life-history at- the examples explored below,g(t) p fZ(t) , where f is the tributes of organisms with complex life cycles, such as the average number of events per season (disturbance fre- Disturbance Regimes and Life-History Evolution 527 quency) and Z(t) is a normal distribution with mean dis- a T Ϫ T turbance date f (disturbance timing) and standard devi- CR if T ! T , ()T RC p C ation j (disturbance predictability). The adult stage is C(TR) (5) assumed to occur after metamorphosis so that T p {0 otherwise, E T ϩ r, where r is the duration of the metamorphic stage, a fixed quantity. Because u1 is evaluated through TE, meta- where TC is the upper time limit for reproduction and a morphosing individuals risk mortality from disturbances is a parameter that controls how rapidly this time con- 1 even though no growth occurs during this stage. Meta- straint approaches. For a 0 C(TR) decreases as TC ap- morphosis thus entails a fixed cost. For organisms that do proaches (earlier offspring are more valuable than later p not undergo metamorphosis,r 0 . ones), and adults are unable to reproduce after TC. The probability of offspring surviving j disturbances af- p Ϫ j ter reproduction isOj (1 l) . The probability of j disturbances occurring from the time of reproduction to the Arithmetic Mean Fitness end of the disturbance season is For an individual that matures at time T in a season where i disturbances occur before adulthood and j dis- Ϫ uej u2 p p 2 turbances occur after reproduction, fitness is Wij(T) Pj ,(2)# # # j! SijO E(W(T)) C(T R). When disturbances are fine grained and occur in a patch dynamic setting so that dis- p ∫TZ turbances happen independently in each patch according whereu 2 TR g(t)dt and TZ is some date well beyond the r to g(t) and the progeny from all patches mix to form a end of the disturbance season (g(t) 0 at TZ). Reproduc- tion occurs after metamorphosis and the adult life stage single breeding population, the arithmetic mean is the p ϩ appropriate way to calculate long-term fitness. Arithmetic (y, a fixed quantity) are completed, so that TR T r ϩ y. mean (AM) fitness is estimated by summing fitness across all possible disturbance seasons weighted by their probability of occurring:

Growth and Reproduction ϱϱ p ͸͸ FAM(T) PPWij ij (6) The growth rate of individuals is assumed to follow a ip0 jp0 logistic form: ϱ Ϫ uei u1 p ͸ (1 Ϫ l)i 1 ip0 i!

dw w ϱ Ϫ p rw 1 Ϫ ,(3) uej u2 dt() k # ͸ (1 Ϫ l)(7)j 2 jp0 j! where k is maximum body mass, w is body mass, and r # E # C is a growth rate constant. The number of offspring pro- Ϫlu Ϫlu p e 12# e # E # C.(8) duced by an individual, E, is related to body mass: Timing of reproduction is optimized by maximizing ﬁtness a(W Ϫ W )ifb W 1 W , E(W) p { CC(4) as a function of T. This is done by taking the ﬁrst derivative 0 otherwise, of FAM with respect to T and setting it equal to 0. First, taking the natural log of both sides for convenience, where W is body mass at time of reproduction, WC denotes p Ϫ Ϫ ϩ ϩ the minimum body size for offspring production, b is a ln[FAM(T)] lu 1lu 2 ln(E) ln(C), (9) parameter that controls the shape of the relationship, and d ln[F (T)] a is a scale factor that adjusts for units of measurement. AM p Ϫlg(T ) dT E EC ϩ lg(T ) ϩϩ p 0, (10) Seasonal Time Constraints R EC

As in the Rowe and Ludwig model, seasonal time con- EC l[g(T ) Ϫ g(T )] ϩϩ p 0. (11) straints on reproduction affect the expected value, or con- REEC tribution, per offspring: 528 The American Naturalist

After incorporating the biological assumptions outlined above (see appendix), equation (11) becomes

W brW 1 Ϫ a ()k [Z(T ) Ϫ Z(T )] p Ϫ .(12) lf RE Ϫ Ϫ TCRTWW C

The left side of equation (12) represents the relative change in fitness due to disturbance regime. When f orl r 0 or r TERT , this side approaches 0, and disturbances have no effect on optimal metamorphosis strategy. Additionally, when the left side of equation (12) is 0 and r and y r r 0,TR T and equation (12) collapses to that of Rowe and Ludwig (1991), where metamorphosis into the adult stage is driven primarily by seasonal time constraints (for cases Figure 1: Optimal body size (W) versus optimal timing of metamor- where their(w) r 0) . The Rowe and Ludwig (1991) m phosis (T) when disturbances cause high mortality (l p 1 ) or none at single-habitat growth model is, therefore, nested within all (l p 0 ), under arithmetic mean assumptions. Shaded region repre- the more general case described here. sents disturbance timing Z(T) with parametersf p 150 andj p 25 . p p p p p Other parameter values:f 1 ,a 0.1 ,b 1 ,k 10 ,WC 2 , p p p p y 10,r 10 ,TC 250 , andr 0.01 . Only individuals that have Disturbance Model Results attained large body sizes (neark p 10 ) early in the season metamorphose during period A. For thel p 1 curve, no metamorphosis occurs during General Behavior period B becauseW 1 k , the maximum body size. Individuals metamorphose during period C because reproduction (at timeT ϩ r ϩ y ) After substituting parameter values, equation (12) can be will occur after the peak of the disturbance season. During period D, W solved for W in terms of T, yielding two roots. Figure 1 increases as the disturbance season wanes, then decreases during period shows the optimal relationship between body size and tim- E in response to TC. ing of metamorphosis for the positive root and parameter valuesf p 1 ,f p 150 ,j p 25 ,a p 0.1 ,b p 1 , will be placed in the disturbance-prone habitat just as the p p p p p p probability of disturbance begins to decline. Very small k 10,WCC2 ,y 10 ,r 10 ,T 250 , and r 0.01. In the case where disturbances do not cause mortality individuals (W ! 6 in this example) risk the worst of the (l p 0 curve), W declines solely as a function of the end disturbance season to continue juvenile growth. After the peak of the disturbance season has passed (period D), the of the season (TC). Body mass, W, becomes progressively marginal benefits of remaining in the juvenile stage begin smaller as TC approaches, and all individuals larger than to outweigh the risks, and smaller individuals that have WC begin metamorphosing with sufficient time remaining survived thus far are expected to continue growing. Finally, to complete their metamorphic and adult stages before TC. This is a state-dependent strategy because the decision to during period E, the approach of TC causes any remaining continue growing or begin metamorphosis is based on individuals to begin metamorphosis at progressively current body mass (Rowe and Ludwig 1991; Nylin and smaller body sizes. Gotthard 1998). When disturbances cause mortality, this pattern is al- Population Synchrony tered in several ways (lp1 curve). Although individuals attaining large body sizes early in the season may meta- In addition to influencing W, disturbances may also affect morphose at this time (period A), no metamorphosis oc- the synchrony of metamorphosis within a population. Fig- curs as disturbances become more likely (period B, where ure 2 shows a series of juvenile growth trajectories. It is W 1 k, the maximum body size). This occurs because off- assumed that growth trajectories are offset because of var- spring produced at this time have low value since they iability early in the life cycle (different oviposition dates, would be placed in the disturbance-prone habitat at the different initial growth rates, etc.). Metamorphosis occurs height of the disturbance season. The largest individuals where growth trajectories cross the W(T) curve. Lack of (W near 10) begin metamorphosis during period C, with metamorphosis during period B causes a relatively large progressively smaller individuals metamorphosing as the number of trajectories to stack up; these trajectories in- disturbance season builds. Note that individuals meta- tersect the steep W(T) curve in period C, producing syn- morphosing at this time will escape the peak of the dis- chronous metamorphosis over a relatively short period of turbance season by becoming adults, and their offspring time. For smaller individuals, a similar delay (period D) Disturbance Regimes and Life-History Evolution 529

Differentiating with respect to T and setting this quantity equal to 0,

EC ln(1 Ϫ l)g(T ) Ϫ ln(1 Ϫ l)g(T ) ϩϩ p 0, (16) EREC

1 EC ln [g(T ) Ϫ g(T )] ϩϩ p 0. (17) ()1 Ϫ l REEC

Comparing equation (17) with equation (11) demonstrates that the optimality conditions for the geometric and arithmetic mean fitnesses differ by a single term: l in the arithmetic mean (AM) model becomesln[1/(1 Ϫ l)] in the GM model. Thus, using the geometric rather than the Figure 2: Effect of the W(T) curve on hypothetical growth trajectories. arithmetic mean influences only how disturbance severity Individuals metamorphose (denoted by arrows) when their growth tra- affects the optimal life-history strategy. Disturbance se- jectories intersect the curve. Dashed line is the mean disturbance date f. verity has a proportionately higher effect under GM as- Parameter values same as those in figure 1;l p 1 curve. Most meta- sumptions becauseln[1/(1 Ϫ l)] 1 l . For l near 0, both morphosis occurs when the W(T) curve declines steeply during periods models will produce essentially the same results, but for C and E (see text). l 1 0.8, ln [1/(1 Ϫ l)] is greater than twice l.Asl approaches its maximum at 1,ln[1/(1 Ϫ l)] approaches in- and synchronous period of metamorphosis (period E) oc- finity. Figure 3 shows that, unlike the AM model, under cur before TC. These results suggest that disturbances can serve to synchronize the metamorphosis of individuals that are following different growth trajectories. Given a particular level of initial within-population size variability, disturbance regimes should favor greater temporal synchrony in metamorphosis while simultaneously increasing the observed variability in body size at metamorphosis.

Geometric Mean Fitness When disturbances are synchronized across patches or when the entire population experiences the same large- scale disturbances, the geometric mean is the appropriate measure of long-term fitness. Geometric mean (GM) fitness is calculated by summing the logarithm of fitness across all possible disturbance seasons:

ϱϱ p ͸͸ FGM(T) PPijln(W ij)(13) ip0 jp0

ϱ Ϫ uei u1 p ͸ l n [(1 Ϫ l)]i 1 ip0 i!

ϱ Ϫ uej u2 ϩ ͸ l n [(1 Ϫ l)]j 2 (14) jp0 j!

ϩ ln(E) ϩ ln(C) Figure 3: Effect of disturbance severity on optimal body size and optimal p Ϫ ϩ Ϫ timing of metamorphosis, under arithmetic mean (top panel) and geo- ln(1 l)u12ln(1 l)u metric mean (bottom panel) assumptions. Parameter values same as those ϩ ϩ in ﬁgure 1. For small l, both models make similar predictions. Large l ln(E) ln(C). (15) causes disturbances to have a much more pronounced effect under geometric mean assumptions. 530 The American Naturalist

phosis before the mean date of disturbance (ﬁg. 5). A similar pattern occurs when frequency is constant but severity is allowed to vary. Under AM assumptions, the relationship between disturbance frequency and severity is multiplicative. For this reason, an increase in frequency can counteract a decrease in severity and vice versa, although severity is bounded between 0 and 1, while frequency can theoretically take on any positive value.

Sensitivity Analysis According to the model, how predictable must the timing of disturbances be to induce changes in an organism’s life- history strategy? Similarly, how frequently must distur- Figure 4: Effect of disturbance predictability (j)ontheW(T) curve, bances recur to produce a change in an organism’s optimal under arithmetic mean assumptions. Dashed line is the mean disturbance maturation strategy? There are no universal answers to p date. Parameter values same as those in figure 1;l 1 curve. When these questions because any answer depends on many in- disturbance regimes are predictable (j p 3 curve), individuals emerge synchronously and at a wide range of body sizes. Unpredictable distur- itial parameter values. Qualitatively, however, the sensitiv- bance regimes produce no life-history response (j p 60 curve, which is ity of the disturbance model predictions to changes in only very similar tol p 0 curve in fig. 1). one or two parameters can be investigated. The difference between a “baseline” curve where no disturbances occur GM assumptions, disturbances drive nearly all the body and a curve where disturbances do occur can be quantified size pattern when l is high. as the sum of squared differences between the two curves. The sum of squares can be interpreted as a measure of disturbance regime selection strength because it describes Disturbance Predictability the degree of difference in optimal phenotype attribut- Variance in the mean date of disturbance (j) also has an able solely to disturbance regime. For individuals in effect on optimal body size at metamorphosis. When dis- disturbance-prone habitats, the greater the departure from turbances always occur within a narrow time interval, the this optimum the lower the expected fitness. optimal strategy is to metamorphose immediately before Figure 6 shows the relationship between selection ϩ the mean date of disturbance, irrespective of body size strength (log 1 scale) and variation in disturbance pre- (j p 3 ; fig. 4). This strategy causes body size at meta- dictability (j) for organisms with different growth rates. morphosis to decline sharply immediately before the mean If body size at maturity is assumed to be fixed, growth date of disturbance, which produces highly synchronous rate can be equated with life span; higher growth rates metamorphosis into the adult stage at a wide range of body sizes. Conversely, when disturbances are unpredictable, individuals do not respond to the disturbance regime, even though disturbances can produce high mortality (j p 60 ; fig. 4). In fact as j increases, the W(T) curve becomes identical to the case where disturbances cause no mortality (l p 0 curve; fig. 1). In this situation, there is simply no life-history strategy, in terms of age and size at metamorphosis, that can be used to avoid a highly unpredictable source of mortality. The sensitivity of this result to changes in j is explored below.

Disturbance Frequency and Severity The mean number of disturbances per season (f) and the Figure 5: Effect of disturbance frequency (f)ontheW(T) curve, under expected mortality from a disturbance (l) also inﬂuence arithmetic mean assumptions. Parameter values same as those in ﬁgure size at and timing of metamorphosis. More frequent dis- 1. Greater disturbance frequencies favor metamorphosis before the mean turbances produce a greater decline in size at metamor- date of disturbance (dashed line), irrespective of body size. Disturbance Regimes and Life-History Evolution 531

predictions. In general, the model showed that life-history strategies that mitigate the negative ﬁtness effects of disturbances are possible, and these strategies are strongly inﬂuenced by disturbance timing, predictability, frequency, and severity.

Disturbance Regimes Can Produce Multiple Seasonal Declines in Body Size at Metamorphosis The disturbance model predicts that when disturbances are sufficiently predictable (i.e., j is low relative to the organism’s life span) and when within-population variance in juvenile body size is large, a single population will metamorphose during two distinct periods. The first period is associated with the disturbance regime and the second with seasonal constraints. Multiple peaks of emergence during Figure 6: Selection strength versus disturbance predictability for organ- a single season have been observed in aquatic insect species isms with different growth rates, under arithmetic mean assumptions. (Vannote and Sweeney 1980; Peckarsky et al. 1993; Moreira Selection strength measures the degree of difference between the no- disturbance curve (l p 0 curve) and a curve where disturbance is a and Peckarsky 1994; Taylor et al. 1998), but this pattern factor (l 1 0 ). Parameter values same as those in figure 1. has been attributed to multi- or semivoltinism of populations (i.e., two distinct cohorts were thought to have suggest a shorter life span, and slower growth rates suggest been observed). Peckarsky et al. (2001) suggested that pat- a longer life span. Sum of squares differences were cal- terns of body size at emergence observed in a mayfly with culated at daily intervals over the course of the entire two consecutive cohorts per season were in fact driven by p p season (fromt 0 toTC 230 ). In all cases, selection a biotic disturbance regime, the onset of fish predation. was strongest when disturbances were more predictable Since multivoltinism alone can produce two consecutive (low j), but the relative magnitude of selection depended emergence groups in the absence of disturbance, care must on the growth rate of the organism. Within the realm of be taken in attributing this type of pattern entirely to a more predictable disturbances (j ! 40 d in this example), disturbance regime when interpreting field data. From a selection strength roughly doubled with every order of modeling perspective, however, the size structure of the magnitude increase in growth rate. As disturbances became population makes no difference in terms of predicting unpredictable, selection strength approached 0 regardless of growth rate, but this happened at much lower values of j for faster growing organisms. For this reason, slower- growing organisms were more likely to show evolutionary responses to highly unpredictable disturbance regimes than faster-growing organisms. Selection strength changed in response to disturbance frequency (shown in fig. 7 as return interval, the reciprocal of f) in a way qualitatively similar to predictability. In all cases, selection strength was highest for short return intervals, but as with disturbance predictability, selection strength was greatest for organisms with slow growth rates. Under GM assumptions, selection strength was proportionately higher at all return intervals, causing the curves in figures 6 and 7 to shift upward.

Discussion Figure 7: Selection strength versus disturbance return interval for organisms with different growth rates, under arithmetic mean assumptions. Combining disturbance ecology with a model of life- Selection strength measures the degree of difference between the no- history evolution generated predictions that are congruent disturbance curve and any curve where disturbance occurs. Parameter with well-known theory and data, as well as some novel values same as those in figure 1. 532 The American Naturalist patterns of body size at metamorphosis, and so the dis- of metamorphosis and growth rate (e.g., Abrams et al. turbance model should work just as well if multiple co- 1996). horts are present. Coarse-Grained and Fine-Grained Disturbance Regimes Produce Similar Life Histories Except When Disturbance Regimes Can Increase Synchrony of Disturbances Are Severe Metamorphosis While Simultaneously Increasing Variation in Body The modeling results show that, when disturbance severity Size at Metamorphosis is low or moderate, using either the geometric or arithmetic mean to estimate fitness results in nearly the same Previous theory (Ludwig and Rowe 1990; Rowe and Lud- life history. This is true regardless of disturbance frequency, wig 1991; Rowe et al. 1994) has shown that time con- predictability, or timing. Thus, population structure, rel- straints can cause organisms to mature over a discrete ative to the spatial scale of disturbances, does not influence period of time, producing a decline in body size at meta- life-history strategies when disturbances cause low or morphosis. Assuming initial variation in body size within moderate mortality. Severe coarse-grained disturbances, a population and constant growth rates, time constraints however, have a much stronger effect on life-history strat- produce an inverse relationship between variance in time egies than do severe fine-grained disturbances. In fact, of metamorphosis and variance in body size at metamor- when disturbances are coarse-grained and severe, they can phosis (illustrated in fig. 2). Disturbance regimes generate completely override the effects due to seasonal time con- a similar phenomenon. Disturbance regimes that are suf- straints. In these cases, metamorphosis should coincide ficiently predictable, cause sufficient mortality, and occur entirely with the disturbance regime, and only the smallest with sufficient frequency can cause variance in time of individuals (those below the minimum size for reproduc- metamorphosis to decrease while variance in body size at tion) should risk disturbances by remaining in the juvenile metamorphosis increases. The degree of this inverse re- growth habitat. lationship depends on the steepness of the W(T) curve and the amount of size variation initially present in the Slow-Growing Organisms Adapt to Disturbance Regimes population. More Readily than Fast-Growing Organisms While size variation within populations is common, constant growth rates during development are not always Life histories of organisms that have fast growth and ma- observed (Gotthard et al. 1999). The distinction between ture quickly do not respond to disturbances as strongly as development rate and growth rate is important here. The slow-growing, long-lived organisms. This finding is con- disturbance model, like most other models of optimal size gruent with results from earlier studies, which suggest that at and timing of metamorphosis, allows development rate disturbances must recur on a timescale comparable to the to vary in response to disturbances or seasonal constraints, organism’s life span to elicit an evolutionary response meaning individuals can accelerate ontogeny to mature (Harper 1977; Lacey et al. 1983; Venable and Brown 1988; earlier. Increasing growth rate involves the accelerated ac- Turner et al. 1998). This occurs because when disturbance quisition of resources, often by way of increased foraging are unpredictable and organisms have rapid growth rates, rates (Lima and Dill 1990). Models that allow variable the best strategy is always to continue the rapid growth, growth rates predict increased growth rates at the expense even if disturbances are frequent and cause high mortality. of increasingly risky foraging behavior (Houston et al. It makes sense that rare or benign disturbances should 1993; Werner and Anholt 1993; Abrams and Rowe 1996). have few consequences for adaptive evolution, but there Experimental manipulations on several insect taxa (refer- is no discrete threshold frequency or severity at which this enced in Johansson and Rowe 1999) have shown that time occurs. Instead, selection strength drops off exponentially constraints accelerate development, which causes individ- with increasing disturbance return interval, and this re- uals to mature earlier and at smaller body sizes. This ob- lationship depends strongly on the growth rate of the servation suggests that flexibility in development rate is organism. an important variable, although growth rates could also Several assumptions made in the model development play a role. To model how disturbances affect life histories may affect the generality of the results presented here. where the assumption of constant growth rates may be These include assumptions of logistic growth of individ- violated, as in some insects (Gotthard et al. 1999) and uals, mortality from disturbances only, and no size- amphibians (Wilbur 1987; Semlitsch and Wilbur 1988), dependent mortality. Equation (3) assumes a fixed upper disturbance regimes need to be incorporated into more limit to growth, and it is possible that nonasymptotic sophisticated models that simultaneously optimize timing growth forms might produce different results. Although Disturbance Regimes and Life-History Evolution 533 the choice of growth curve can influence optimal strategies winter frost or the drying date of temporary rain pools, (Day and Taylor 1997; Czarnołe˛ski and Kozłowski 1998), phenomena that are well described by probability dis- it is important to note that disturbance regimes cause tributions. metamorphosis even during the exponential phase of Although the disturbance model developed here treats growth trajectories (fig. 2). Thus, although an unrestricted a specific kind of evolution (evolution of optimal body maximum body size could result in a larger size at met- size and timing of metamorphosis), some of the model’s amorphosis for some individuals (and, thus, increase pop- qualitative results may apply to other evolutionary and ulation-wide variance in body size at metamorphosis), dis- ecological scenarios. For example, the disturbance mod- turbance should produce at least some decline in body eling approach could be used to locate the critical range size for even nonasymptotic growth functions. Distur- of disturbance frequencies within which organisms are bances in the juvenile growth habitat were assumed to be likely to adapt to particular disturbance regimes, such as the only source of mortality in order to highlight their plants adapting to fire regimes (Christensen 1985). Beyond effects on life-history strategies. Mortality in the adult hab- this critical range, disturbances may actually exclude taxa itat would likely diminish the fitness benefits of maturing, from the system (an ecological effect) rather than drive resulting in larger final body sizes and flattening of the evolutionary change of life-history attributes. Broadening body size/timing of metamorphosis reaction norm (Rowe this approach to include other aspects of disturbance (spa- and Ludwig 1991; Werner and Anholt 1993). Thus, the tial extent, synergistic effects of multiple kinds of distur- effects of juvenile-habitat disturbance on patterns of body bance) and other evolutionary processes (behavioral, size at metamorphosis should be reduced as adult mor- morphological) could clarify under which conditions dis- tality increases. For organisms with size-specific mortality turbance regimes are important for ecology, evolution, or (e.g., John 1964), disturbance regimes should favor meta- both. morphosis at smaller body sizes if mortality risk from disturbances increases with body size. This would occur because while survivorship for growing individuals de- Acknowledgments creases, survivorship for their small offspring would be relatively high. The reverse would be true if mortality risk I thank P. Abrams, C. Caudill, J. Dahl, N. Hairston, Jr., J. from disturbances decreases with body size. Hoekstra, T. Kawecki, K. Macneale, B. Peckarsky, K. Reeve, Following the example of previous models, the time L. Rowe, and two anonymous reviewers for comments on constraint on reproduction was included in the distur- an earlier version of this manuscript. C. McCulloch and bance model as a continuous function that decreases to 0 L. Stirling Churchman gave advice on model development. at TC, the last day of the reproductive season. In fact, the One of the reviewers provided valuable insights on gen- end of the reproductive season could also be treated as a eralizing the model to include both the arithmetic and type of disturbance, with a characteristic frequency, timing, geometric mean cases. This project was conducted with severity, and predictability. This approach would be par- support from the Cornell University Departments of En- ticularly suitable for season-ending events, such as the first tomology and Ecology & Evolutionary Biology.

APPENDIX

Incorporating Biological Assumptions

From equation (11), optimal body size at metamorphosis under the arithmetic mean assumptions is given by

dE(W(T)) 1 dC(T )1 Ϫ ϩϩR p l[g(TRE) g(T )] 0. (A1) dT E(W(T)) dT C(TR)

TheE /E term represents the relative gain in ﬁtness, as a function of T. By the chain rule of calculus and by substituting equation (3),

dE(W(T)) dE dw dE W pprW 1 Ϫ .(A2) dT dW dt dW() k

Under the assumption in equation (4), 534 The American Naturalist

Ϫ W brW ()1 k dE(W(T)) 1 when W 1 W , p W Ϫ W C (A3) { C dT E(W(T)) 0 otherwise.

TheC /C term represents the relative change in offspring value due to seasonal time constraints on reproduction. Under the assumption in equation (5),

dC(T )1 R # p Ϫ a Ϫ .(A4) dT C(TRCR) T T

Assuming thatg(t) p fZ(t) , where f is disturbance frequency and Z(t) is a probability distribution describing the timing of disturbance events within a season (assumed later to be a normal distribution), equation (A1) becomes

W brW 1 Ϫ a ()k [Z(T ) Ϫ Z(T )] p Ϫ .(A5) lf RE Ϫ Ϫ TCRTWW C

The geometric mean form (eq. [17]) can be treated in the same way, and the result differs only in the l term.

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